A spin-1 particle with charge $q$ and magnetic moment $\vec{\mu}=\frac{gq}{2mc}\vec{S}$ is situated in a magnetic field $B=B_0\vec{z}$. At $t=0$ the particle is found in an eigenstate $\hat{S}_y$ with eigenvalue $+\hbar$. Determine the state of the particle at some time $t$. What is the probability that a measurement of $S_y$ at time $t$ will yield $\hbar$? In the $\hat{S}_z$ eigenbasis, two of the eigenstates of $S_y$ are
$$|+\hbar\rangle =\frac{1}{2} \left( \begin{array}{ccc}
1 \\
i\sqrt{2} \\
-1 \end{array} \right)
,
|0\rangle=\frac{1}{\sqrt{2}} \left( \begin{array}{ccc}
1 \\
0 \\
1 \end{array} \right) $$
I suspect that the information about the eigenstates of $S_y$ provided at the end are only pertinent to the probability question. So far, I have written down that the eigenstates of the Hamiltonian are the same as those of $\hat{S}_z$. Thus, since $\hat{H}=-\vec{\mu}\cdot\vec{B}=\frac{-gq}{2mc}\hat{S}_zB_0$,
$$\hat{H}|+z\rangle=\frac{-gq}{2mc}B_0\hbar |+z\rangle$$
$$\hat{H}|-z\rangle=\frac{gq}{2mc}B_0\hbar |-z\rangle$$
Then, I wrote down $|\psi(0)\rangle=|+\hbar\rangle$. I am stuck on how to write $\hbar$ in terms of the z basis elements (if that is the correct next step). Could someone give me a few pointers in how to continue this problem, and how I would start finding the probabilities for which they ask?
Best Answer
The fact that the eigenstates of the Hamiltonian are those of $S_z$, is indeed apparent due to the fact that the two commute.
The key fact that you should notice is that your initial state isn't an eigenstate of the Hamiltonian, and therefore it must change in time. In your case is will oscillate.
Regarding your initial state, you haven't used all the info the question offered. Think what is the meaning of the $+\hbar$ eigenvalue at the initial state. You can determine the initial state by it.
After you figure this out, use the evolution operator:
$$ \hat{U}(t)=e^{-i\hat{H}t/\hbar} $$
To figure out how does your state evolve with time